Problem 41 - A Complete Resource Book in Mathematics for JEE Main

Question

The equations of the straight lines passing through \((-2,-7)\) and cutting an intercept of length three units between the straight lines \(4 x+3 y=12\) and \(4 x+3 y=\) 3 are (A) \(x+2=0, y+7=\frac{7}{24}(x+2)\) (B) \(x-2=0, y+7=-\frac{7}{24}(x+2)\) (C) \(x+2=0, y+7=-\frac{7}{24}(x+2)\) (D) \(x+2=0, y+7=-\frac{7}{12}(x+2)\)

Step-by-Step Solution

Answer

The correct answer is (C): \(x+2=0, y+7=-\frac{7}{24}(x+2)\).

Step 1: Understanding the Problem

We need to find the equations of lines that pass through the point \((-2, -7)\) and intersect the lines \(4x + 3y = 12\) and \(4x + 3y = 3\) such that the intercepted segment is 3 units long. We also have four options to check if they solve the given problem.

Step 2: Find the Intersection Points

To find where a line intersects \(4x + 3y = 12\) and \(4x + 3y = 3\), we substitute the equation of a line passing through \((-2, -7)\). If \(y - 7 = m(x + 2)\) is such a line, then substituting in \(4x + 3y = k\), we solve for the x-coordinate and then find the point on this respective line by substituting for y.

Step 3: Calculate the Length of the Intercept

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). We need this length to be 3, where \((x_1, y_1)\) is the point of intersection with \(4x + 3y = 12\) and \((x_2, y_2)\) with \(4x + 3y = 3\).

Step 4: Check the Options

Substitute the point \((-2, -7)\) in each of the given line equations and check if they satisfy passing through this point. Use the resulting line equations to find the values of \(m\) and verify which ones produce an intercept length of 3 between the given lines.

Step 5: Verification Through Calculation

For each valid line, calculate the specific intersection points by plugging \(4x + 3y = 12\) and \(4x + 3y = 3\) into the line equation derived using given point \((-2, -7)\). Calculate the lengths using the distance formula and check which option yields 3.

Key Concepts

Intersection of Lines

To find where two lines intersect, all you need to do is solve their equations simultaneously. This means finding a pair of values that satisfies both equations at the same time.
For instance, given two lines with equations, such as \(4x + 3y = 12\) and \(4x + 3y = 3\), finding their intersection with another line involves substituting the equation of this other line into each of these equations.

• Let's say the equation of the line passing through a point \((-2, -7)\) is \(y + 7 = m(x + 2)\).
• To find the intersection with \(4x + 3y = 12\), replace \(y\) in the equation with \(m(x + 2) - 7\) and solve for \(x\).
• Do the same substitution for \(4x + 3y = 3\) to find the second point of intersection.

By solving these, you can locate the specific points where each line crosses, helping to determine the intercepts.

Distance Formula

The distance formula is essential when you need to find out how far apart two points are on a plane. The formula is derived from the Pythagorean theorem and works like magic for pinpointing distances:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula calculates the distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\).

• Use this formula to determine the length of the segment intercepted between two intersecting lines.
• First, find the two intersection points as discussed earlier.
• Substitute the coordinates of these intersection points into the distance formula to calculate the segment length.

In this exercise, it's crucial to achieve a result of exactly 3 units to match the problem requirements.

Equation of a Line

An equation of a line in a 2D plane is typically expressed in the form \(y = mx + c\), where:

• \(m\) is the slope of the line, indicating how steep the line is.
• \(c\) is the y-intercept, showing where the line crosses the y-axis.

For a line through a specific point \((x_0, y_0)\), use the point-slope formula:
\[y - y_0 = m(x - x_0)\]To find the needed line equations for this exercise:

• Identify the slope \(m\) through calculations involving known intersecting lines.
• Substitute \((-2, -7)\) into the point-slope formula to create the line equation.
• This approach ensures the derived line passes through the required point.

By following these steps, you can confirm which line fits the criteria given.